Volume 2009, Article ID 781579, 9 pagesdoi:10.1155/2009/781579 Research Article An Exponentially Fitted Method for Singularly Perturbed Delay Differential Equations Fevzi Erdogan Departm
Trang 1Volume 2009, Article ID 781579, 9 pages
doi:10.1155/2009/781579
Research Article
An Exponentially Fitted Method for Singularly
Perturbed Delay Differential Equations
Fevzi Erdogan
Department of Mathematics, Faculty of Sciences, Yuzuncu Yil University, 65080 Van, Turkey
Correspondence should be addressed to Fevzi Erdogan,ferdogan@yyu.edu.tr
Received 4 November 2008; Accepted 16 January 2009
Recommended by Istvan Gyori
This paper deals with singularly perturbed initial value problem for linear first-order delay differential equation An exponentially fitted difference scheme is constructed in an equidistant mesh, which gives first-order uniform convergence in the discrete maximum norm The difference scheme is shown to be uniformly convergent to the continuous solution with respect to the perturbation parameter A numerical example is solved using the presented method and compared the computed result with exact solution of the problem
Copyrightq 2009 Fevzi Erdogan This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
Delay differential equations play an important role in the mathematical modelling of various practical phenomena in the biosciences and control theory Any system involving a feedback control will almost always involve time delays These arise because a finite time is required
to sense information and then react to it A singularly perturbed delay differential equation
is an ordinary differential equation in which the highest derivative is multiplied by a small parameter and involving at least one delay term1 4 Such problems arise frequently in the mathematical modelling of various practical phenomena, for example, in the modelling of several physical and biological phenomena like the optically bistable devices5, description
of the human pupil-light reflex6, a variety of models for physiological processes or diseases and variational problems in control theory where they provide the best, and in many cases the only realistic simulation of the observed phenomena7
It is well known that standard discretization methods for solving singular perturbation
problems are unstable and fail to give accurate results when the perturbation parameter ε is
small Therefore, it is important to develop suitable numerical methods to these problem,
whose accuracy does not depend on the parameter value ε, that is, methods that are
uniformly convergent with respect to the perturbation parameter8 10 One of the simplest
Trang 2ways to derive such methods consists of using an exponentially fitted difference scheme see, e.g.,10 for motivation for this type of mesh, which are constructed a priori and depend
of the parameter ε, the problem data and the number of corresponding mesh points In
the direction of numerical treatment for first-order singularly perturbed delay differential equations, several can be seen in4,7,11
In order to construct parameter-uniform numerical methods, two different techniques are applied Firstly, the numerical methods of exponential fitting type fitting operators
see 9, which have coefficients of exponential type adapted to the singular perturbation problems Secondly, the special mesh approach see 11, 12, which constructs meshes adapted to the solution of the problem
In the works of Amiraliyev and Erdogan11, special meshes Shishkin mesh have been used The method that we propose in this paper uses exponential fitting schemes, which have coefficients of exponential type
2 Statement of the Problem
Consider a model problem for the initial value problems for singularly perturbed delay differential equations with delay in the interval I 0,T:
εut atut btut − r ft, t ∈ I,
where I 0, T m
p1I p , I p {t : r p−1< t ≤ r p }, 1 ≤ p ≤ m and r s sr, for 0 ≤ s ≤ m and I0
−r, 0 for simplicity we suppose that T/r is integer 0 < ε ≤ 1 is the perturbation parameter,
a t ≥ α > 0, bt, ft, and ϕt are given sufficiently smooth functions satisfying certain regularity conditions to be specified and r is a constant delay The solution ut displays in general boundary layers at the right side of each points t r s 0 ≤ s ≤ m for small values
of ε.
In this paper, we present the completely exponentially fitted difference scheme on the uniform mesh The difference scheme is constructed by the method of integral identities with the use of exponentially basis functions and interpolating quadrature rules with weight and remainder terms integral form10 This method of approximation has the advantage that the schemes can also be effective in the case when the continuous problem is considered under certain restrictions
In the present paper, we analyze a fitted difference scheme on a uniform mesh for the numerical solution of the problem 2.1 In Section 2, we describe the problem In
Section 3, we state some important properties of the exact solution InSection 4, we construct
a numerical scheme for solving the initial value problem2.1 based on an exponentially fitted difference scheme on a uniform mesh InSection 5, we present the error analysis for approximate solution Uniform convergence is proved in the discrete maximum norm A numerical example in comparison with their exact solution is being presented inSection 6 The approach to construct discrete problem and error analysis for approximate solution is similar to those ones from10,11
Notation Throughout the paper, C will denote a generic positive constant possibly subscripted that is independent of ε and of the mesh Note that C is not necessarily the same at each occurrence
Trang 33 The Continuous Problem
Here, we show some properties of the solution of2.1, which are needed in later sections for the analysis of appropriate numerical solution Let, for any continuous function g,g ∞,I
denotes a continuous maximum norm on the corresponding interval
Lemma 3.1 Let a, b, f ∈ C1I, ϕ ∈ C1I0 Then, for the solution ut of the problem 2.1 the
following estimates hold
u t
where
C1 α−1f ∞,I1 1 α−1b ∞,I1ϕ ∞,I0,
C p α−1f ∞,I p 1 α−1b ∞,I pC p−1, p 2, 3, , m. 3.2
Proof see11
4 Discretization and Mesh
In this section, we construct a numerical scheme for solving the initial value problem2.1 based upon an exponential fitting on a uniform mesh
We denote by ω N0the uniform mesh on I:
ω N0
t i iτ, i 0, 1, 2, , N0; τ r
N , pN N0
which contains N mesh points at each subinterval I p 1 ≤ p ≤ m:
ω N,pt i:p − 1N 1 ≤ i ≤ pN , 1≤ p ≤ m, 4.2 and consequently
ω N0 m
p1
To simplify the notation, we set g i gt i for any function gt, and moreover y i denotes an approximation of ut at t i For any mesh function{w i } defined on N0, we use
w t,i w i − w i−1
τ ,
Trang 4The approach of generating difference methods through integral identity
χ i τ−1
t i
t i−1
Lu tψ i tdt χ i τ−1
t i
t i−1
with the exponential basis functions
ψ i t exp −a i
t i − t
ε
, t i−1≤ t ≤ t i , 4.6
where
χ i τ−1
t i
t i−1
ψ i tdt
−1
a i ρ
1− exp− a i ρ , ρ τ
We note that function ψ i t is the solution of the problem
−εψ
i t a i ψ i t 0, t i−1≤ t < t i ,
ψ i
t i
The relation4.5 is rewritten as
χ i τ−1ε
t i
t i−1
utψ i tdt a i χ i τ−1
t i
t i−1
u tψ i tdt b i χ i τ−1
t i
t i−1
u t − rψ i tdt R i f i , 4.9
with the remainder term
R i R1i R2i R3i ,
R1i χ i τ−1
ti
t i−1
a t − at iu tψ i tdt,
R3i χ i τ−1
ti
t i−1
b t − bt iu t − rψ i tdt,
R3i χ i τ−1
ti
t i−1
f t i − ftψ i tdt.
4.10
Taking into account4.5 and using interpolating rules with the weight see 10, we obtain the following relations:
εθ i u t,i a i u i b i u i −N R i f i , 1≤ i ≤ N0, 4.11
Trang 5θ i 1 χ i τ−1a i ε−1
t i
t i−1
t − t i
and a simple calculation gives us
θ i a i ρ
1− exp− a i ρ exp− a i ρ
As a consequence of the 4.11, we propose the following difference scheme for approximation2.1:
Ly i: εθi y t,i a i y i b i y i −N f i , 1≤ i ≤ N0,
where θ iis defined by4.13
5 Analysis of the Method
To investigate the convergence of the method, note that the error function z i y i − u i, 0≤ i ≤
N0, is the solution of the discrete problem
εθ i z t,i a i z i b i z i −N R i , 1≤ i ≤ N0,
where R i and θ iare given by4.10 and 4.13, respectively
Lemma 5.1 Let y i be approximate solution of 2.1 Then the following estimate holds
y ∞,ω N,p ≤ ϕ ∞,ω N,0 Q p α−1p
k1
f ∞,ω N,k Q p −k , 1≤ p ≤ m, 5.2
where
Q p −k
⎧
⎪
⎪
p
s k 1
1 α−1b ∞,I s, for 0 ≤ k ≤ p − 1. 5.3
Proof The proof follows easily by induction in p.
Trang 6Lemma 5.2 Let z i be solution of 5.1 Then following estimate holds
p
k1
Proof It evidently follows from5.2 by taking ϕ ≡ 0 and f ≡ R.
following estimate holds
Proof To this end, it su ffices to establish that the functions R k i k 1, 2, 3, involved in the expression for R i, admit the estimate
R k∞,ω N ,p ≤ Cτ, k 1, 2, 3. 5.6 Using the mean value theorem, we get
a t − a
t i aξt − t i ,
max
ω N,p
aξ t − t i ≤ Cτ, ξ ∈ t i−1, t i 1
Hence
R1
i ≤ Cττ−1 t i
t i−1
u t ψ i tdt, 5.8
and taking also into account that 0≤ ψ i t ≤ 1 and usingLemma 3.1, we have
R1
For R2i , in view of b ∈ C1I and usingLemma 3.1, we obtain
R2
i ≤ τ−1 t i
t i−1
b t − b
t i
u t − r ψ i tdt ≤ C t i
t i−1
u ξ − r dξ. 5.10
Hence
R2
∞,ω N ,p ≤ C
t i
t
u ξ − r dξ, 5.11
Trang 7and after replacing s ξ − r this reduces to
R2
∞,ω N ,p ≤ C
t i −r
t i−1 −r
u s ds C 0
−r
ϕ s ds t i
t i−1
u s ds, 5.12
which yields
R2∞,ω
N ,p ≤ Cτϕ 1,0 C p
The same estimate is obtained for R3i in the similar manner as above
Combining the previous lemmas we get the following final estimate, that is, uniformly convergent estimate
Theorem 5.4 Let u be the solution of 2.1 and y be the solution of 4.14 Then the following
estimate holds
y − u ∞,ω N,p ≤ Cτ, 1 ≤ p ≤ m. 5.14
6 Numerical Results
We begin with an example from Driver2 for which we possess the exact solution
εut ut ut − 1, t ∈ 0, T,
The exact solution for 0≤ t ≤ 2 is given by
u t
⎧
⎪
⎪
−1 − 2ε t 1 εe −t/ε
ε−1
ε 1 1
ε
t
We define the computed parameter-uniform maximum error e N,p ε as follows:
e N,p ε y − u ∞,ω N,p , p 1, 2, 6.3
where y is the numerical approximation to u for various values of N, ε We also define the computed parameter-uniform convergence rates for each N:
r N,p lne N,p /e 2N,p
The values of ε for which we solve the test problem are ε 2 −i , i 1, 2, , 8.
Trang 8Table 1: Maximum errors e N,1
ε and convergence rates r N,1 on ω N,1
ε N 128 N 256 N 512 N 1024 N 2048
2−1 0.0033688 0.0016866 0.000843849 0.000422062 0.000211065
2−2 0.00381473 0.00191236 0.000957428 0.000479026 0.000239591
2−3 0.00386427 0.00194230 0.000973693 0.000487882 0.000243900
2−4 0.00382489 0.00193278 0.000971476 0.00048701 0.000243823
2−5 0.00374366 0.00191245 0.000966391 0.000485738 0.000243505
2−6 0.00358208 0.00187183 0.000956223 0.000433195 0.000242869
2−7 0.00326581 0.00179104 0.000935915 0.000477811 0.000241598
2−8 0.00268346 0.0016329 0.00895519 0.00467957 0.000239057
Table 2: Maximum errors e N,2
ε and convergence rates r N,2 on ω N,2.
ε N 128 N 256 N 512 N 1024 N 2048
2−1 0.00319858 0.00164347 0.000832995 0.000419339 0.000211065
2−2 0.00600293 0.00300639 0.00150442 0.000752515 0.000376334
2−3 0.00780800 0.00396966 0.00200100 0.00100461 0.000503328
2−4 0.0185227 0.00951902 0.00482057 0.00242576 0.001216820
2−5 0.0388137 0.0202932 0.0103797 0.00525228 0.002641280
2−6 0.0747962 0.0405973 0.0211784 0.0108201 0.005461600
2−7 0.131822 0.0765885 0.0414891 0.0216210 0.011040200
2−8 0.149561 0.133579 0.0774847 0.0419350 0.021842300
These convergence rates are increasing as N increases for any fixed ε Tables1and2
thus verify the ε-uniform convergence of the numerical solutions and the computed rates are
in agreement with our theoretical analysis
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... theoretical analysis Trang 91 R Bellman and K L Cooke, Differential-Difference Equations, Academic Press,... of ε for which we solve the test problem are ε −i , i 1, 2, , 8.
Trang 8Table... dξ, 5.11
Trang 7and after replacing s ξ − r this reduces to
R2